MAST90068 (Category theory)

Fri, Jul 15, 2016

This is the class webpage for the masters level course MAST90068: Groups, Categories & Homological Algebra. All classes are in Evan Williams, see the timetable for more details. Office hours will be on Tuesdays 12:30-2pm and Thursdays 10:30am-12pm, in Room 168.

Assessment

Up to 40 pages of assignments (30%: three assignments worth 10% each, due early, mid and late in semester), a 3-hour written examination (70%, in the examination period).

In every lecture there will be assigned exercises (see the lecture notes posted below) and the three graded assignments will be largely made up of questions from those exercises. Ergo, keep up with the exercises and you’ll be in good shape for the assessment.

Lecture notes

All lecture notes will be posted here before class:

  • Lecture 1: Categories
  • Lecture 2: Functors
  • Lecture 3: Examples of functors, universal problems
  • Lecture 4: Examples of functors, the simplex category
  • Lecture 5: Examples of functors, simplicial sets
  • Lecture 6: The Yoneda lemma
  • Lecture 7: Graphs versus categories
  • Lecture 8: Cones and Limits
  • Lecture 9: Examples of limits and colimits
  • Lecture 10: Adjunctions
  • Lecture 11: Beginnings of Homological algebra, complexes and their cohomology
  • Lecture 12: Singular homology of topological spaces
  • Lecture 13: The long exact sequence in homology
  • Lecture 14: The snake lemma (Hilton & Stammbach)
  • Lecture 15: Injective, projective and free objects (Hilton & Stammbach)
  • Lecture 16: Extensions of modules
  • Lecture 17: Ext via projective presentations
  • Lecture 18: Examples of Ext and long exact sequences
  • Lecture 19: Injective abelian groups
  • Lecture 20: There are enough injectives
  • Lecture 21: Injective envelopes and Matlis’ theorem
  • Lecture 22: Homotopy and projective resolutions. This lecture covered p.5 to p.8 of the attached lecture notes, up to the beginning of Section 3. Note that we are working only in the category of modules so A = RMod and B = SMod for some rings R,S. There is only one thing that I defined in the lecture which is not present in these notes, which is the concept of an additive functor from RMod to SMod. This is simply a functor for which the functions between Hom sets are all morphisms of abelian groups.
  • Lecture 23: Left derived functors, essentially just the definition and independence of resolutions from p.8 of my notes on “Derived Functors”. We also discussed basic properties of flat modules, for which see this extract from Hilton & Stammbach.
  • Lecture 24: Tor, the left derived functor of the tensor product.
  • Lecture 25: Long exact sequence of left derived functors. We covered most of Section 7.1 of the notes. The naturality of the connecting homomorphism (in Theorem 34) was stated but the proof was left as an exercise, the same is true of Proposition 37 and Proposition 38 (I stated but did not prove them). Finally we discussed that the natural transformations induced on derived functors by Definition 9 are used to make Tor a bifunctor, for which see p.2 of these notes.
  • Lecture 26: Right derived functors and Ext. We covered Section 7.2 of the notes plus the argument that the new derived functor definition of Ext agrees with the old one.

As we head into the final week of the subject, I recommend that you take a look at the introduction to Gelfand and Manin’s book “Homological algebra” and this MathOverflow post by Buchsbaum himself explaining the historical development of some of these important theorems.

Besides these notes, you can find useful stuff elsewhere on this website.

Assignments

Assignment 1 due Tuesday 23rd August in class:

  • Lecture 2 Exercises 4, 5, 9, 11.
  • Lecture 3 Exercise 2.
  • Lecture 5 Exercise 3, 4.
  • Lecture 6 Exercises 1, 2.

Some remarks on Assignment 1.

Assignment 2 due Tuesday 4th October in class:

  • Lecture 10 Exercise 2.
  • Lecture 11 Exercises 5,6.
  • Lecture 12 Exercise 1.
  • Lecture 13 Exercises 1,2.
  • Lecture 16 Exercise 3.
  • Lecture 17 Exercises (from Hilton & Stammbach p.94) 2.1, 2.2, 2.3, 2.5, 2.6, 2.9.

A comment on the assignment questions from p.94 of Hilton & Stammbach. Question 2.5 asks you to check that the given formula makes E(A,B) into an abelian group. Q2.6 asks you to check that the functoriality in the first variable is linear with respect to this group structure. Q2.9 asks you to compute it in an example. This computation needs to be justified. You can justify it however you like, but the two obvious methods are: compute the group structure directly from the definition of Q2.5, OR prove that the isomorphism E(A,B) = Ext(A,B) that we discussed matches the natural abelian group structure on Ext(A,B) with the “new” structure on E(A,B) from Q2.5, and then use the (easy) calculation of Ext(A,B) from lectures.

Assignment 3 due Thursday 20th October in class:

  • Lecture 18, Exercises (from Hilton & Stammbach p.106) 5.7(ii), 5.8.
  • Lecture 19, Exercises (from Hilton & Stammbach p.33) 7.1, 7.5.
  • Lecture 20, Exercises 1, 2.
  • Lecture 21, Exercise 4.

Some notes. For Lecture 21, Exercise 4 you should prove both that Z_{p^\infty} is injective and that the canonical map from Z_p is an essential monomorphism. In Exercise 7.5 of Lecture 19, you should read “abelian group” for “group”.

Here are partial solutions to Assignment 3, specifically to Exercise 5.8 from Hilton & Stammbach.

Exam

The bulk of the exam will be taken from the exercises assigned in lectures. To be clear, this may include exercises that were not assigned for any homework assignment. In addition to the exercises there will be at least one proof from the lectures that you will be asked to give in the exam. These will be taken from the following (short) list:

  • Lecture 2, Lemma on p.2 and Theorem on p.3.
  • Lecture 8, Lemma on p.4.
  • Lecture 10, Theorem on p.3 (all parts).
  • Lecture 14, Lemma 5.1 (the Snake lemma).
  • Lecture 18, the proof that Ext(Z_r,Zq) = Z{(r,q)} given there.
  • Lecture 24, Lemma on p.2, Examples on p.5.
  • Lecture 27, Lemma on p.2.

Here is a list of excluded exercises that are guaranteed to not appear on the exam:

  • Lecture 4, Exercise 2.
  • Lecture 5, Exercise 2.
  • Lecture 21, Exercises 1, 2.

Finally, note that bulk does not mean entirety.

References

For the category theory component:

  • Borceux “Handbook of Categorical Algebra I
  • Mitchell “Theory of Categories
  • Schubert “Categories
  • Adamek, Herrlich, Strecker “Abstract and concrete categories
  • Maclane “Categories for the working mathematician
  • Murfet “Foundations for category theory” (FCT)
  • Grothendieck “Introduction to EGA1” explains the importance of the Yoneda lemma and categories in modern algebraic geometry.

For the homological algebra:

  • Hilton & Stammbach “A course in homological algebra”
  • Weibel “An Introduction to homological algebra”

Other: